Some Results On Waring Problem And Estimates Of Maximal Divisor Of K-TH Power

In this dissertation, which contains two chapters. We study Waring-Goldbach problem in the set of Piatetski-Shapiro prime and estimates of maximal divisor of k-th power in short intervals.In chapter 1, we study the number of solutions of the equation with prime variables:in the set of Piatetski-Shapiro prime, where s > 5, N > 0 is & sufficiently large integer. Combining with Heath-Brown's identity, we can convert it into bounding the corresponding exponential sums. Thus we only need to estimate "type I" sum and "type II" sum. The aim of this chapter is to improve the result of Zhai wenguang [8]. And we obtain the following theorem:Suppose are fixed real number, 0 < ri < 1, satisfyingThenwhere T(N; s) denotes the number of solutions of the equation with prime variables:Then we arrive at the following results:(l)For any fixed, every sufficiently large N = 5 (mod 24) can be written as sums of five squares of primes, of which every prime have the form of [n1/r]-(2)For any fixed , every sufficiently large N = 5(mod24) can be written as sums of five squares of primes, of which one prime has the form of [n1/r].In chapter 2, we study . We want to find y as small aspossible, such that the above sum in short intervals has an asympotic formula. Where max is & fixed real number, x > 0 is a sufficiently large number, 0 < y = o(x). Using Perron formula, we obtain the main term of the above sum. Moreover, by the convolution method, we can convert the estimates of the error term into estimating the error term of , which is . Where In this chapter, we arrive at the following result: Suppose, wheree > 0 is a arbitrary small constant, and then...